For now, assume that P != NP. Is there a "complete" class of problems between P and NP-hard, and if so, what is it called? The two key words here are
By between, I mean that it runs in super-polynomial time (not in P), but sub-exponential time (NP-hard is believed to take exponential time or greater).
By complete, I mean that reductions exist. E.g. just like how NP-complete problems like TSP and 3-SAT can be reduced to each other, the "in-between" class of problems similarly should have reductions to each other.
As an example, integer-factorization is widely believed not to be NP-hard, but not to be in P either, so it is a perfect candidate for this class. But is there a wide range of problems that can be reduced to/from integer factorization to create a "complete" class? Or are the problems between P and NP-hard mostly disconnected?